Symmetrize a polynomial forgetting the commutativity property of multiplication

Question

I need a script to authomatically symmetrize a given polynomial. For example, if the input is

xy

the output should be

(xy+yx)/2

The same principle should work also for higher order polynomial. For example, if the input is

xyz

the output should be

(xyz+xzy+yxz+yzx+zxy+zyx)/6

The input is, in general, a polynomial. If the input is

xy+wz

the output should be

(xy+yx+wz+zw)/2

It may happen that some terms come with some powers. If the input is

x^2 z

the output should be either

(xxz + xxz +xzx+ xzx + zxx + zxx)/6

or

(2x^2z+2xzx+2zx^2)/6

Both these ouputs are good. Thanks in advance for the help.

Are you always dealing with monomials, or should a solution expect things like x y + w z? — J. M.'s missing motivation, Commented Oct 14, 2018 at 16:03
The input could be a polynomial as well. For example, if the input is $xy+wz$, the output should be $(xy+yx+wz+zw)/2$, but I know a priori what are the variables that come into play. Moreover, of course, it may appen to have things like $x^2y$ as input. In this case, the output should be either $(xxz+xzx+xxz+xzx+zxx+zxx)/6$ or $(2x^2z + 2xzx+ 2zx^2)/6$. Both outputs are ok for me. — AndreaPaco, Commented Oct 14, 2018 at 16:08
@J.M.iscomputer-less, I've edited the question in order to meke my question more precise. — AndreaPaco, Commented Oct 14, 2018 at 16:19

Henrik Schumacher · Accepted Answer · 2018-10-14 16:28:01Z

4

What about this?

f[NonCommutativeMultiply[x__]] := Mean[Permutations[NonCommutativeMultiply[x]]]

f[x ** y ** z]

1/6 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x + z ** x ** y + z ** y ** x)

Adding

f[x_Times] := f /@ x
f[x_Plus] := f /@ x
f[x_?NumericQ] := x

allows us to treat also polynomials:

f[3 x ** x ** y + y ** y ** y + 54 x ** z ** y]

x ** x ** y + x ** y ** x + y ** x ** x + y ** y ** y + 9 (x ** y ** z + x ** z ** y + y ** x ** z + y ** z ** x + z ** x ** y + z ** y ** x)

answered Oct 14, 2018 at 16:13

Henrik Schumacher

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$\begingroup$ Thanks for your help. Can it handle a generic polynomial $P(x,y,z)$ such as $P(x,y,z)=3x^2y+y^3+54xzy$? $\endgroup$
– AndreaPaco
Commented Oct 14, 2018 at 16:22
$\begingroup$ Thanks a lot for the update. You gave me a huge help. Just a curiosity more. In the input the non-commutative multiplication $**$ is already made explicit. Is there a way to authomatically turn a traditional multiplication (such as $x*y$) into a non-commutative one, (such as $x**y$)? $\endgroup$
– AndreaPaco
Commented Oct 14, 2018 at 16:32
1

$\begingroup$ Yes, you can always tuse replacement rules, e.g. x x y //. {Power[a_, k_Integer?Positive] :> NonCommutativeMultiply @@ ConstantArray[a, k], Times -> NonCommutativeMultiply} (however this won't work out with the coefficients). $\endgroup$
– Henrik Schumacher
Commented Oct 14, 2018 at 16:45

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